what does the minus sign mean when we say that Inductive Reactance = - Capacitive Reactance in a resonance circuit? sure there is no negative ohm value, is it indicative of the characteristics of the circuit for the direction of current...blah blah blah..

what does the minus sign mean when we say that Inductive Reactance = - Capacitive Reactance in a resonance circuit? sure there is no negative ohm value, is it indicative of the characteristics of the circuit for the direction of current...blah blah blah..

thx

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When we speak of reactance and use either of the two formulas:

Code ( (Unknown Language)):

X[SIZE=1]C[/SIZE] = 1 / (2*∏*f*C)

X[SIZE=1]L[/SIZE] = 2*∏*f*L

The units are ohms and they are positive real numbers.

When we wish to combine a resistance and a reactance we have to multiply an inductive reactance by j, and a capacitative reactance by (1/j). If you know the rules of complex algebra you remember that:

Code ( (Unknown Language)):

1/j = -j

and that is where the minus sign comes from. So if an impedance has a minus sign in front of the reactive part it is not to be interpreted a s negative ohms because when we take the magnitude of the impedance we square the real and imaginary parts, and take the square root giving us a poasitive number.

We used Cartesian (?) coordinates (good ol' graph paper). The +X axis was for resistance, and because the Inductive voltage was 90* ahead of the resistive, it was plotted against the +Y axis, and because the Capacitive voltage lagged by 90* it was plotted against the -Y axis. (Teacher: "Trust me, it works!")

The Inductive reactance and the capacitive reactance were summed, and if I recollect, we used the Pythagorean theorem to calculate the total impedance. -- I might be wrong about this paragraph; it's been over 20 years since I needed it: I wound up going into the digital area.

We used Cartesian (?) coordinates (good ol' graph paper). The +X axis was for resistance, and because the Inductive voltage was 90* ahead of the resistive, it was plotted against the +Y axis, and because the Capacitive voltage lagged by 90* it was plotted against the -Y axis. (Teacher: "Trust me, it works!")

The Inductive reactance and the capacitive reactance were summed, and if I recollect, we used the Pythagorean theorem to calculate the total impedance. -- I might be wrong about this paragraph; it's been over 20 years since I needed it: I wound up going into the digital area.

--Rich

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You have the ticket mate. Another way to think about the imaginary unit is as a rotation operator. If the real axis (+X axis) is assigned to 0° then a multiplication of a reactance in ohms by j, the imaginary unit, is equivalent to a rotation of a value plotted on the real axis by +90°, and a multiplication by -j is equivalent to a rotation of a value plotted on the real axis by -90°.

Let me drive home one more point. There is nothing imaginary about AC circuit values or measurements. The use of complex numbers is a convenience that keeps track of the algebra describing the real behavior of actual things you can touch and feel. It is equally valid to use polar notation which replaces real and imaginary parts with magnitude and phase. The rules of phasor algebra do not require the use of complex numbers and they give the identical results.

ELI the ICE man is your friend. The sign of the value tells you which direction the current phase is shifted relative to the voltage. In an inductor, the voltage leads the current by 90 degrees, so it's +j. In a capacitor, the voltage LAGS the current by 90 degrees, so it's -j. Generally speaking voltages are more commonly measured than currents in reactive circuits, hence the particular convention of inductance being + and capacitance being -. In fact, for parallel circuits, it's sometimes convenient to swap the + and -j operators.

J is what is referred to as an imaginary number. It is the square root of negitive one. Intuitively you would think that such a think can not exist, but it does and is very useful for describing complex filter circuits (absolutely necessary as a matter of fact). I tend to think of it as a sign, similar to + and -, but with rules of its own.